A Nonparametric Riemannian Framework for ProcessingHigh Angular Resolution Diffusion Images (HARDI)

Alvina Goh1 Christophe Lenglet2 Paul M. Thompson3 Rene Vidal11Johns Hopkins University

{agoh,rvidal}@jhu.edu2University of Minnesota

clenglet@umn.edu

3University of California, Los Angeles

thompson@loni.ucla.edu

AbstractHigh angular resolution diffusion imaging has become an

important magnetic resonance technique for in vivo imaging.Most current research in this field focuses on developingmethods for computing the orientation distribution function(ODF), which is the probability distribution function of wa-ter molecule diffusion along any angle on the sphere. Inthis paper, we present a Riemannian framework to carry outcomputations on an ODF field. The proposed frameworkdoes not require that the ODFs be represented by any fixedparameterization, such as a mixture of von Mises-Fisherdistributions or a spherical harmonic expansion. Instead,we use a non-parametric representation of the ODF, and ex-ploit the fact that under the square-root re-parameterization,the space of ODFs forms a Riemannian manifold, namelythe unit Hilbert sphere. Specifically, we use Riemannianoperations to perform various geometric data processingalgorithms, such as interpolation, convolution and linearand nonlinear filtering. We illustrate these concepts withnumerical experiments on synthetic and real datasets.

1. IntroductionDiffusion magnetic resonance imaging (MRI) is an imag-

ing technique that produces in vivo images of biological

tissues by exploiting the constrained diffusion properties

of water molecules. Water diffusion is hindered less along

fibrous structures as the molecules encounter fewer tissue

barriers when they move in that direction. The signal inten-

sity at each position depends on the local microstructure in

which the water molecules diffuse as well as the strength and

direction of the applied magnetic field gradients. Multiple

MR scans with varying gradient directions and strengths are

used to capture the complete diffusion profile of a tissue. As

the directions of maximum diffusion indicate the structural

anisotropy of the medium, diffusion MRI can be used to infer

the organization and orientation of tissue components. Exper-

iments have shown that water diffusion is indeed anisotropic

in organized tissues such as the spinal cord, muscles, heart

or brain white matter. This has generated much enthusiasm

and high expectations, because diffusion MRI is presently

the only available approach to non-invasively study the three-

dimensional architecture of human tissues, and quantify their

physical and geometrical properties.

Several techniques can be used to reconstruct orienta-

tion distribution functions from diffusion MRI. One classi-

cal technique is known as diffusion tensor imaging (DTI)

[4]. In DTI, the diffusivity profile is characterized by a

single oriented 3D Gaussian probability distribution func-

tion. Water diffusion is then represented mathematically

with a symmetric positive semi-definite (SPSD) tensor field

D : R3 → SPSD(3) ⊂ R

3×3 that measures the extent of

diffusion in a direction v ∈ R3 as v�Dv. Even though DTI

assumes a relatively simple diffusion model, it has been a

successful imaging technique in regions of the brain and

spinal cord with significant white-matter coherence. For

instance, DTI has enabled the mapping of anatomical con-

nections in the central nervous system [5, 16]. However,

DTI models the diffusion with a single tensor, so it can only

reveal a single fiber orientation within each voxel. When

fibers cross or bifurcate within a voxel, which is common

in complex brain structures, DTI will not be able to provide

accurate insights into the diffusivity profile or fiber geometry.

Recent advances in diffusion MRI address this well-

known limitation of DTI. In particular, an imaging

method known as high angular resolution diffusion imaging

(HARDI) has been proposed [30]. HARDI measures diffu-

sion along n uniformly distributed directions on the sphere

and is able to characterize complex fiber geometries. Sev-

eral reconstruction techniques can be used to characterize

diffusion based on the obtained HARDI signal. A common

approach is to construct the orientation distribution function

(ODF) from these measurements. The ODF is a probabilitydistribution function that quantifies the probability of water

diffusion along different directions on the sphere. A signifi-

cant amount of the current research on HARDI focuses on

computing the ODF from HARDI signals. One of the earliest

methods uses the Funk-Radon transform to estimate ODFs

[31]. In addition, spherical harmonic expansions have also

been used to approximate ODFs [9, 12, 24]. Such methods

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are typically very fast as the ODF is computed analytically.

Finally, higher-order tensors, which can leverage the work

done in DTI, have also been used to model diffusivity [3, 14].

While much of the current research efforts focus on com-

puting the ODFs efficiently and accurately, much less work

has been done on the processing of these high-dimensional

datasets. That is, given a set of ODFs, how would one per-

form statistical and principal components analysis, or basic

operations such as interpolation, filtering and convolution?

Unlike in DTI where the geometry of the SPSD space is well

studied and several frameworks [2, 18, 21, 25] have been

proposed, such frameworks are less common for HARDI

data. In order to perform any operations, one will first need a

metric to compare different ODFs. As in DTI, this is an open

issues and several choices, such as Kullback-Leibler diver-

gence (KLD) [10], Jensen-Shannon divergence (JSD) [8] and

Riemannian distance between a mixture of von Mises-Fisher

distributions [22], have been proposed. KLD is difficult to

compute for the spherical data with high angular resolution

and it also has to be symmetrized. Both the symmetrized

version of KLD (SKLD) and SJD do not obey the triangular

inequality and therefore are not metrics. The work of [22]

presents a Riemannian metric and a possible framework for

HARDI processing. In their work, the precomputed ODF

is represented by a mixture of von Mises-Fisher distribu-

tions, where the mixture parameters are computed using an

expectation-maximization algorithm. Once that is done, the

authors show that it is possible to compute distances and

various Riemannian operations between different mixtures

of von Mises-Fisher distributions using closed form formu-

lae. However, this framework suffers from two major issues.

First, the number of components in the mixture model is ar-

bitrarily chosen. To find the optimal number of components

that one should use, the difficult question of model selection

must be addressed. The second issue is that a nonlinear

least-squares technique is used to compute the unknown pa-

rameters. This technique may not be stable as the number

of parameters increases, when the complexity of the tissue

increases. Therefore, a framework is needed to allow the

processing of ODFs without having to resort to such models,

while respecting their intrinsic mathematical properties.

Paper Contributions: In this paper, we present a Rieman-

nian framework for the processing of ODFs. We operate

directly on the precomputed ODF and do not require that the

ODF be represented by any parametric model such as a mix-

ture of von Mises-Fisher distributions. We exploit the fact

that ODFs are probability density functions defined on the 2-

sphere S2, and that under the square-root re-parametrization,

the space of ODFs forms a Riemannian manifold, namely

the unit Hilbert sphere. Therefore, various Riemannian op-

erations such as the exponential map, logarithmic map and

geodesics are not only available in closed form, but also

easily and efficiently computed. We use these operations

to introduce various geometric data processing algorithms

such as interpolation, convolution, and linear and nonlinear

filtering. We present experiments on synthetic and real data.

2. The Riemannian Manifold of OrientationDistribution Functions

In this section, we show that the orientation distribution

functions lie on a Riemannian manifold. We first present

a brief summary of the theory of Riemannian manifolds.

We then illustrate the Riemannian structure of the space of

ODFs, under the square-root representation, which gives

closed-form expressions for various Riemannian operations.

2.1. Review of Riemannian Manifolds

A differentiable manifold M of dimension d is a topolog-

ical space that is homeomorphic to the Euclidean space Rd.

The tangent space TxM at x is the vector space that contains

the tangent vectors to all 1-D curves on M passing through

x. A Riemannian metric on a manifold M is a bilinear form

that associates to each point x ∈ M, a differentiable varying

inner product 〈·, ·〉x on the tangent space TxM at x. The

norm of a vector v ∈ TxM is denoted by ‖v‖2x = 〈v,v〉x.

The Riemannian distance dist(xi,xj) between two points

xi and xj lying in the manifold is defined as the minimum

length over all possible smooth curves on the manifold be-

tween xi and xj . The geodesic curve from xi to xj , γ, is

the smooth curve with minimum length.

Given a tangent vector v ∈ TxM, there exists a unique

geodesic γv(t) starting at x with initial velocity v, and

this geodesic has constant speed equal to ‖v‖x. The ex-ponential map, expx : TxM → M maps a tangent vec-

tor v to the point on the manifold that is reached at time

1 by the geodesic γv(t). The inverse of expx is the log-arithm map and denoted by logx : M → TxM. For

two points xi and xj on the manifold M, the tangent

vector to the geodesic curve from xi to xj is defined as

v = −−→xixj = logxi(xj), and the exponential map takes v to

the point xj =expxi(logxi

(xj)). In addition, γv(0) = xi

and γv(1) = xj . The Riemannian distance between xi

and xj is defined as dist(xi,xj) = ‖ logxi(xj)‖xi . Linear

geodesic interpolation makes use of the exponential and loga-

rithm maps and is defined as x = expxi(w−−→xixj), w ∈ [0, 1].

x is the linear interpolation at t = w of xi and xj . Finally,

we recall that the Riemannian metric, exponential and log-

arithm maps depend on the point x under consideration,

hence the subscripts reflecting this dependency.

2.2. Parameterization of ODFs

We will now show how to impose a Riemannian structure

on the space of ODFs or equivalently, the space of probabil-

ity density functions (PDF). In particular, we will adopt a

“spherical” version of the Fisher-Rao metric that allows for

closed-form expressions of the various Riemannian opera-

tions. The class of constrained non-negative functions under

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study here is the set of probability density functions on the

2-sphere S2. Specifically, we consider the set of PDFs

P = {p : S2 → R+|∀s ∈ S2,p(s) ≥ 0;

∑s∈S2

p(s) = 1}.

The question of how to equip the space of PDFs with a

differential manifold structure, a Riemannian metric and a

family of affine connections has a long history. Nevertheless,

it remains an active and important research area. Treating

statistical quantities as geometric structures has the advan-

tage of preserving the invariance of such structures under

change of coordinates. Studying probability and information

via differential geometry is known as information geometry.

The reader is referred to [1] for a complete description. Rao

[27] first introduced the Riemannian structure formed by the

statistical manifold whose elements are probability density

functions. In addition, he also showed [27] that the Fisher-Rao metric determines a Riemannian metric. The Fisher-Rao

metric is defined as

〈qj , qk〉pi=∑s∈S2

qj(s)qk(s)1

pi(s), (1)

where qj , qk ∈ TpiP are tangent vectors and Tpi

P is the

set containing the functions tangent to P at the point pi.

This metric was later shown to be the unique intrinsic met-ric on the statistical manifold in [7], therefore invariant to

re-parameterizations (essentially coordinate transforms) of

the functions. However, this representation turns out to be

extremely difficult to work with as the computation of the

geodesic between two elements is not easy [29].

Even though the space P turns out to be difficult to work

with, we know that it is not the only possible representation

for PDFs. There are many different re-parameterizations

of PDFs that are equivalent. These include the cumulative

distribution function, log density function and square-root

density function. Each of these parameterizations will lead

to a different resulting manifold. Depending on the choice of

representation, the resulting Riemannian structure can have

varying degrees of complexity and numerical techniques

may be required to compute geodesics on the manifold. For

example, the authors of [23] chose the log density represen-

tation and used a shooting technique to find geodesics on this

space. However, this space has a complicated Riemannian

structure and the numerical method used in [23] sometimes

leads to large errors. Therefore, a natural question to ask

is: Is it possible to use a re-parameterization such that the

resulting manifold is simple and the Riemannian operations

are easy to compute, preferably in closed-form? Once an ef-

ficient representation is found, the corresponding Fisher-Rao

metric, which depends on the tangent vector, can be used as

the Riemannian metric.

In this paper, the choice of parameterization is the square-

root representation. This representation has been used in

[15, 20, 29]. We will consider the space of ODFs, which

are discrete multi-variate probability density functions. The

main reason behind choosing the square-root representation

is that the resulting manifold is a unit sphere in a Hilbert

space with the Fisher-Rao metric being the usual L2 met-

ric. Therefore, the various Riemannian operations such as

geodesics, exponential maps, logarithmic maps are available

in closed form.

The square-root density function is one of the most effi-

cient representations found as of today and is defined as

ψ(s) =√

p(s),where ψ(s) is assumed to be non-negative to ensure unique-

ness. The space of such functions is defined as:

Ψ = {ψ : S2 → R+|∀s ∈ S2,ψ(s) ≥ 0;

∑s∈S2

ψ2(s) = 1}.

(2)

From Eq. (2), it is easy to see that the functions ψ lie on

the positive orthant of a unit Hilbert sphere1. In addition,

Ψ forms a convex subset of the unit Hilbert sphere. The

advantage of choosing the square-root density becomes im-

mediately obvious, as many of the Riemannian expressions

for the unit Hilbert sphere are well-known and closed-form

expressions. Now, making use of the representation in Eq. (2)

and the fact that Riemannian metrics are determined up to

a constant scaling factor, we can rewrite Eq. (1) and obtain

the Fisher-Rao metric as

〈φj ,φk〉ψi=∑s∈S2

φj(s)φk(s),

where φj ,φk ∈ TψiΨ are tangent vectors.

2.3. Geodesic Marching on the Space Ψ

In this section, we will show how to compute the various

Riemannian operations in Ψ in order to evolve (or “march”)

along its geodesic curves. For any two functions ψi, ψj ∈Ψ, the geodesic distance between these two points on a unit

Hilbert sphere is simply the angle between them,

dist(ψi,ψj) = ‖ logψi(ψj)‖ψi

= cos−1〈ψi,ψj〉 = cos−1

(∑s∈S2

ψi(s)ψj(s)

), (3)

where 〈·, ·〉 is the normal dot product between points in the

sphere under the L2 metric. From the differential geometry

of the sphere, the exponential map is defined as

expψi(φ) = cos(‖φ‖ψi

)ψi + sin(‖φ‖ψi)

φ

‖φ‖ψi

, (4)

where φ ∈ TψiΨ is a tangent vector at ψi and

‖φ‖ψi=√〈φ,φ〉ψi

.

In order to ensure that the exponential map is bijective and

stays on the positive orthant, we restrict ‖φ‖ψi ∈ [0, π2 ].

1Note that the usage of the word “sphere” might be confusing. ODFs are definedon the 2-sphere S2, whereas in Eq. (2), each ψ is a point on the unit Hilbert sphere.

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The logarithm map from ψi to ψj is then given by

−−−→ψiψj = logψi

(ψj) =ψj − 〈ψi,ψj〉ψi√

1 − 〈ψi,ψj〉2cos−1〈ψi,ψj〉.

(5)

2.4. Statistics on the Space Ψ

We will now briefly illustrate how to calculate the mean

and principal components of points lying on Ψ. As defined

by Frechet in [13], the intrinsic mean ψ is the solution to the

following minimization problem

ψ = arg minψ∈Ψ

1n

n∑i=1

dist(ψ,ψi)2. (6)

Note that, unlike in the Euclidean case, in general there is no

closed form for the intrinsic mean on Riemannian manifolds.

Moreover, there is no guarantee that it exists or is unique.

However, one can prove the existence and uniqueness of

the mean [17] when the manifold has non-positive sectional

curvature or by assuming that the data lies in a small enough

neighborhood. Even though the mean might not be well-

defined for a generic Riemannian manifold (consider two

antipodal points on a sphere for instance), it is unique for

a convex subset of a Riemannian manifold [6]. This is true

despite that the sectional curvature of the unit Hilbert sphere

having a positive value of 1. Therefore, the mean is uniqueand exists for Ψ, the space of ODFs. Similarly to Eq. (6),

the weighted mean is defined as

ψ = arg minψ∈Ψ

n∑i=1

widist(ψ,ψi)2.

where wi ≥ 0 and∑n

i=1 wi = 1, in order to ensure that the

solution exists and is unique. Notice that when wi = 1n , we

have the intrinsic mean. In addition, the (weighted) mean is

characterized by being the unique solution ton∑

i=1

wi logψ(ψi) = 0. (7)

Since the exponential and logarithm maps are known (Eq.

(4) and (5)), it is possible to compute ψ using geodesic

marching, as detailed in Algorithm 1.

Algorithm 1 (Weighted Mean)Given data points ψ1, . . . ,ψn ∈ Ψ, a predefined threshold

ε, maximum number of iterations T ,

1. Initialize t = 1, ψ1 = xi for a random i.

2. While t ≤ T or ‖φ‖ψ ≥ ε,

(a) Compute tangent vector φ =∑n

i=1 wi logψt(ψi),

(b) Set ψt+1 = expψt(φ).

Given ψ, the calculation of principal components on a

Riemannian manifold is not as straightforward as in the

Euclidean case. In [11], it is shown that finding principal

components boils down to doing principal components anal-

ysis (PCA) on the tangent vectors logψ(ψi) ∈ TψΨ about

the mean ψ. This algorithm, known as Principal Geodesic

Analysis (PGA), is summarized in Algorithm 2.

Algorithm 2 (Principal Geodesic Analysis)Given data points ψ1, . . . ,ψn ∈ Ψ,

1. Compute intrinsic mean ψ as in Algorithm 1.

2. Calculate the tangent vectors φi = logψ(ψi)about ψ.

3. Construct the sample covariance matrix cov(ψ) =1n

∑ni=1 φiφ

�i .

4. Perform eigenanalysis of the matrix cov(ψ), with the

eigenvectors {ui}di=1 giving the principal directions.

{ui}di=1 forms an orthonormal basis for TψΨ.

3. Processing of ODFs FieldsIn this section, we show how to process a field of ODFs

using the tools of §2. In particular, we consider operations

such as interpolation, convolution, and filtering of ODFs.

3.1. Spatial Interpolation of ODFs

Interpolation is one of the most important operations in

data processing. In general, almost every geometric transfor-

mation requires interpolation to be performed on an image

or a volume, e.g., translating, rotating, scaling, warping. The

fundamental question that interpolation raises is that of the

estimation of unknown data, by using known data. Stan-

dard interpolation methods are often based on attempts to

generate continuous data from a set of discrete data sam-

ples through an interpolation function. To the best of our

knowledge, spatial interpolation of ODFs is a new issue [8].

While trilinear interpolation has been previously used in [8],

we present a complete overview of how one does spatial

interpolation on both regular and non-regular grids here.

Linear interpolationWe will first start with the simplest case, namely lin-

ear interpolation. Let x1, x2 ∈ R and consider the inter-

val [x1, x2]. Assume we know the ODFs at x1 and x2,

ψ(x1) and ψ(x2) respectively, and we want to estimate

unknown ψ at point x ∈ [x1, x2]. It is well-known that

ψ(x) = ψ(x1) + w−−−−−−−−→ψ(x1)ψ(x2), w = x−x1

x2−x1is the linear

interpolation equation in Euclidean space. Therefore, for

the Riemannian manifold Ψ, we again have the closed-form

expression ψ(x) = expψ(x1)(w−−−−−−−−→ψ(x1)ψ(x2)).

Interpolation on a regular gridWe will now move on to the slightly more sophisticated

processing tools commonly used in image processing: multi-

dimensional interpolation such as bilinear and trilinear inter-

polation. These are extensions of linear interpolation to in-

terpolating functions on a n-dimensional regular grid; n = 2for bilinear and n = 3 for trilinear interpolation. Here, we

2499

will assume that we have known data at the 2n endpoints of

a n-dimensional regular grid ∪ni=1[xi,1, xi,2]. In the standard

Euclidean case, multi-dimensional interpolation is defined

at a point x = [x1, . . . , xn] as

ψ(x) =2n∑

j=1

wj(x)ψ(xj) ⇒2n∑

j=1

wj(x)−−−−−−−→ψ(x)ψ(xj) = 0,

where xj = [(xj)1, . . . , (xj)n] are the coordinates of the

endpoints, ψ(xj) are the corresponding values and wj(x) =∏ni=1

(1 − |xi−(xj)i|

xi,2−xi,1

). For a Riemannian manifold Ψ, the

equivalent expression is

2n∑j=1

wj(x) logψ(x) ψ(xj) = 0.

Notice its similarity to the weighted mean in Eq. (7). There-

fore, the solution is defined as

arg minψ(x)∈Ψ

2n∑j=1

wj(x)dist(ψ(x),ψ(xj))2.

Since interpolation is an operation on a fixed grid, the op-

timization is over ψ and not over the fixed x’s. We use

Algorithm 1 to compute the interpolated values.

Interpolation on a non-regular gridFinally, we consider the case where we have m

known measurements {ψ(xj)}j=1,...m scattered on a non-

regular grid {xj}j=1,...,m. We can either use simple

nearest-neighbor interpolation or we can estimate the un-

known values via the inverse distance weighting (IDW)

method [28]. That is, we define the interpolation as the

solution to arg minψ(x)∈Ψ

∑mj=1 wjdist(ψ(x),ψ(xj))2,

where wj(x) = K(x,xj)−p∑m

i=1 K(x,xj)−p is a simple IDW weighting

function as defined in [28], K is a given metric operator (dis-

tance) from xj to x, and p is a positive real number, called

the power parameter.

3.2. Convolution and Filtering of ODFs

Convolution is the cornerstone of many data processing

applications such as denoising, smoothing, edge detection,

image sharpening and filtering. One of the best known appli-

cations will be that of Gaussian smoothing that is commonly

used to remove noise from data. While this is a simple and

robust filtering technique commonly used in various applica-

tions, it suffers from the problem of blurring and mislocating

discontinuities in the data. These discontinuities usually cor-

respond to important structures of the data, such as edges

and corners in images. As the isotropic Gaussian kernel

is the solution to the linear diffusion equation, Gaussian

convolution is equivalent to linear diffusion. In particular,

Gaussian smoothing corresponds to having the same diffu-

sivity constant c independently of the spatial coordinates x.

Therefore, various nonlinear diffusion filtering techniques,

which smooth the data while preserving discontinuities, have

been proposed. The main idea behind these anisotropic filter-

ing methods is to make use of a non-homogeneous diffusivity

functions c(·) that depend on the data at x. For example, in

[26], a diffusivity function c(·) based on the derivative of the

data at x is used to control the smoothing near the edges of

the image. Thus, diffusion across edges is greatly reduced

while being allowed along edges. This prevents edges and

corners from being smoothed during the filtering process.

In this section, we will first consider the problem of doing

convolution and (isotropic) filtering on a field of ODFs. In

the latter part of this section, we will detail how to perform

anisotropic filtering. The motivation for filtering diffusion

data is that the signal-to-noise ratio in HARDI data is gener-

ally low, especially with the high gradient strengths neces-

sary for imaging and resolving crossing white matter tracts

and other regions of complex tissue architecture.

ConvolutionTraditionally, filtering is based on the idea of convolut-

ing data with different filters g. Discrete convolution of ψwith the function g is defined as ξ(x) = ψ(x) ∗ g(x) =∑

u g(u)ψ(x+u). Assuming that∑

u g(u) = 1 and rewrit-

ing, we get∑

u g(u)−−−−−−−−−−→ξ(x)ψ(x + u) = 0.

When the data lies on Ψ, it is immediate to see that the

equivalent expression is∑

u g(u) logξ(x) ψ(x + u) = 0.

Again, the solution is defined as

arg minξ(x)∈Ψ

∑u

g(u)dist(ξ(x),ψ(x + u))2. (8)

When g(u) ≥ 0, the optimization problem in Eq. (8) remains

convex, and the convergence to the correct solution is again

guaranteed and solved using Algorithm 1. When g(u) < 0,

convergence is guaranteed when the Hessian of the function

in Eq. (8) is positive definite.

Anisotropic FilteringWe will first review the well-known problem of minimiz-

ing the spatial irregularity of a scalar field ψ : Rd → R.

Consider the cost function

E(ψ) =∫

‖∇ψ(x)‖2dx, =∫ d∑

i=1

〈δiψ, δiψ〉dx, (9)

where ∇ψ(x) = [δ1ψ, . . . , δdψ]� is the spatial gradient

of ψ and ∇ = [δ1, . . . , δd]� the gradient operator. We can

compute the gradient of the energy functional by considering

a tangent vector φ(x) at ψ(x) and evaluating the functional

derivative. We have

δφE(ψ) = 2∫〈∇ψ(x),∇φ(x)〉dx. (10)

Now, we want to find the gradient ∇E such that it is inde-

pendent of the tangent vector φ(x). That is to say, we are

trying to find ∇E such that δφE(ψ) =∫ 〈∇E,φ(x)〉dx.

This is done by assuming homogeneous Neumann boundary

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conditions and applying integration by parts. We get

δφE(ψ) = −2∫〈Δψ(x),φ(x)〉dx, (11)

where Δψ(x) =∑d

j=1 δ2j ψ is the Laplacian operator.

Therefore, the gradient of the energy is ∇E = −2Δψ(x).Using gradient descent, this brings us to the familiar evolu-

tion equation

ψt+1(x) = ψt(x) − γ∇E = ψt(x) + 2γΔψ(x), (12)

where γ > 0.

For anisotropic filtering of a scalar field, the cost func-

tion to be minimized is such that it penalizes smoothing

across edges, i.e., when the derivative of the scalar field is

large. Usually, this is done via a penalizing function and

an example of such a function is c(x) = exp(−x2

κ2 ). In that

case, the discrete implementation for anisotropic filtering is

a straightforward variation of Eq. (12), where one penalizes

contribution of Δuψ(x) in the spatial direction u according

to the norm of the gradient ‖δuψ(x)‖ in the direction u.

That is,

∇E = −2c(‖δuψ(x)‖)Δuψ(x). (13)

Now, for a field of ODFs, and making use of the intrinsic

gradient descent scheme, the evolution equation becomes

ψt+1(x) = expψt(x)(−γ∇E)

= expψt(x)(2γΔψ(x)), (14)

where γ > 0, where Δψ(x) is the Laplacian operator on the

space Ψ. Similar to what was done for DTI [19], we define

the equivalent of Eq. (9) as

E(ψ) =∫ d∑

i,j

〈δiψ(x), δjψ(x)〉ψ(x)dx,

where δiψ(x) ∈ Tψ(x)Ψ is the partial derivative of ψ in

direction i. The gradient of the energy is then

∇E = −2∑i,j

Δijψ(x), where Δij = δiδj .

Therefore, to perform the anisotropic filtering of field of

ODFs, we proceed as in Eq. (13), and modify Eq. (14) to

account for the penalizing function c(·).Finally, we will show how to approximate the operator

Δij for a given field of ODFs ψ. To this end, recall that

under the finite central difference methods for functions, we

have the following expressions for the Euclidean derivatives,

with {ei} being the orthonormal vectors of the canonical

basis:

δiψ(x) =1ε(ψ(x +

εei

2) − ψ(x) + ψ(x) − ψ(x − εei

2)),

δ2i ψ(x) =

1ε2

(ψ(x + εei) − 2ψ(x) + ψ(x − εei)),

δiδjψ(x) =1ε2

(ψ(x +εei√

2+

εej√2) − ψ(x +

εei√2− εej√

2)

− ψ(x − εei√2

+εej√

2) + ψ(x − εei√

2− εej√

2)).

Therefore, for the first and second order derivatives in Rie-

mannian space Ψ, we have the following approximations:

δiψ(x) ≈ 1ε(−−−−−−−−−−−−→ψ(x)ψ(x +

εei

2) −

−−−−−−−−−−−−→ψ(x)ψ(x − εei

2)),

δ2i ψ(x) ≈ 1

ε2(−−−−−−−−−−−→ψ(x)ψ(x + εei) +

−−−−−−−−−−−→ψ(x)ψ(x − εei),

δiδjψ(x) ≈ 1ε2

(−−−−−−−−−−−−−−−−−→ψ(x)ψ(x +

εei√2

+εej√

2)

−−−−−−−−−−−−−−−−−−→ψ(x)ψ(x +

εei√2− εej√

2) +

−−−−−−−−−−−−−−−−−→ψ(x)ψ(x − εei√

2− εej√

2))

−−−−−−−−−−−−−−−−−−→ψ(x)ψ(x − εei√

2+

εej√2).

The derivation of the various approximations of the deriva-

tives ψ proceeds as in the DTI case [19, 25].

4. ExperimentsIn this section, we present experiments on synthetic and

real datasets using the proposed framework. We first illus-

trate the various operations previously introduced, such as

interpolation, filtering, and computation of the mean and

principal components (PC) of the synthetic ODF fields in

Fig. 1. Fig. 1(a) shows the bilinear interpolation of the four

endpoints where the original ODFs are shaded in orange.

The top right and bottom left voxels contain ODFs of 1 fiber,

the bottom right ODF of 2 fibers and the top left ODF of

3 fibers, with 1 fiber pointing out of the plane. Next, we

illustrate Gaussian filtering. We add Gaussian noise (in the

Riemannian sense) to the resulting ODF field in Fig. 1(a), as

shown in Fig. 1(b). Fig. 1(c) shows the results of Gaussian

filtering. Notice that Gaussian filtering managed to remove

a significant amount of noise. Figs. 1(d)-1(f) show the Rie-

mannian mean and the first two principal components with

eigenvalues 0.104, 0.041 of the ODF field from Fig. 1(a).

The first PC is oriented from left to right, whereas the second

PC is oriented going out of the plane.

Next, we perform anisotropic filtering (AF) on a ODF

field with a sharp discontinuity, as shown in Fig. 1(g). The

function c(x) = exp(−x2

κ2 ) is used in AF, and 30 iterations

are performed. Again, we add noise to this ODF field, as

shown in Fig. 1(h). Fig. 1(i) shows the results of Euclidean

Anisotropic Filtering (EAF), where instead of taking into

account the Riemannian structure of the ODFs, we simply

treat each ODF as a vector in Euclidean space, and perform

anisotropic filtering in the classical Euclidean way. Fig.

1(j) shows the results of Riemannian Anisotropic Filtering

(RAF). From Figs. 1(i)-1(j), one can see that RAF is able to

respect the discontinuity whereas EAF blurs the edges.

Next, we examine the differences in performance be-

tween RAF and EAF as we vary the amount of noise

added to the ground-truth ODF field over 100 trials for

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(a) Bilinear Interpolation from 4 ODFs (b) Noise corrupted ODF field (c) Riemannian Gaussian Filtering

(d) Mean

(e) 1st PC,

e=0.104

(f) 2nd PC,

e=0.041

(g) Synthetic ODFs used for AF (h) Noise corrupted ODF field for AF (i) Euclidean Anis. Filtering (j) Riemannian Anis. Filtering

Figure 1. Illustration of interpolation, Gaussian filtering, statistics computation, and anisotropic filtering of synthetic ODF fields.

Noise level, σ=10% 5σ 4σ 3σ 2σ σEuclidean measure 0.74 0.44 0.30 0.24 0.20Riemannian measure 0.92 0.78 0.71 0.66 0.63

Table 1. Mean of (Amount of error after RAF)/(Amount of error

after EAF) in 100 trials under two different measures.

5 different noise levels σ − 5σ, where σ = 10%. For

both types of AFs, we measure the amount of error

from the ground-truth ODF field using two measures, Rie-

mannian∑

x ‖ logψtruth(x)(ψfiltered(x))‖x and Euclidean∑x ‖ψtruth(x)−ψfiltered(x)‖ dissimilarity. In each trial,

we iterate AF 30 times. Table 1 shows the ratio of error for

RAF to EAF under the Euclidean and Riemannian measures.

RAF always results in a lesser amount of error compared to

EAF. In addition, as the amount of noise that is added to the

ODF field increases and the discontinuity becomes blurred,

we see that the error in RAF converges to that of EAF.

Finally, we apply our Riemannian framework to a sub-

set of a HARDI human brain database [8] of 12 subjects.

Diffusion weighted MR images were obtained using the fol-

lowing imaging parameters: 21 axial slices (5 mm thick),

TR/TE=6090/91.7 ms, with a 128 × 128 acquisition matrix

(1.8 mm in-plane resolution). 3 with no diffusion sensiti-

zation and n = 27 diffusion weighted images at b = 1132s/mm2 were acquired. Gradient directions were evenly dis-

tributed on the hemisphere. The ODFs of the 12 subjects

were fluidly registered to a target template and transformed

to isotropic voxel resolution (128 × 128 × 93 voxels with

1.7 × 1.7 × 1.7 mm3 resolution) [8]. We will first use our

framework to calculate the mean ODF field of the 12 sub-

jects. A portion of the results in an axial slice near the corpus

callosum are shown in Fig. 2. Next, we apply our Rieman-

nian anisotropic filtering method to the mean ODF and the

ODF fields of the 12 subjects. Fig. 3 shows a portion of the

results of RAF and Fig. 3(a)-3(b) show the resulting ODF

fields after applying RAF to the ODF fields of the calculated

mean and subject 1. Again, the function c(·) used in RAF is

a Gaussian function, and 30 iterations are done. Notice that

RAF is able to smooth the ODF field sufficiently but still

preserves the discontinuities between different bundles.

5. ConclusionsWe present a Riemannian framework for the processing

of ODFs that does not require that the ODF be represented

by any parametric model. The various Riemannian opera-

tions are in closed form and easily and efficiently computed.

Results on synthetic and real data demonstrate and quantify

the advantage of working with our proposed framework as

opposed, for instance, to a Euclidean approach. Choosing

the correct metric for ODFs remains an open question and

we present a rigorous and well founded framework with

desirable properties that the other metrics do not have.

Acknowledgments. We would like to thank Ming-Chang Chiang for pro-

viding us with the registered datasets. Work supported by startup funds

from JHU, by grants NSF CAREER IIS-0447739, NIH R01 HD050735,

NIH R01 EB007813, NIH P41 RR008079, NIH P30 NS057091, ONR

N00014-05-10836 and ONR N00014-09-1-0084.

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(a) Zoomed-out ODF of mean (b) ODF of mean (c) ODF of subject 1 (d) ODF of subject 2

(e) ODF of subject 3 (f) ODF of subject 4 (g) ODF of subject 5 (h) ODF of subject 6

Figure 2. Calculation of mean ODF field from 12 subjects. Fig. 2(a) show the mean ODF field of one slice where the zoomed-in results of

the red box shown in Fig. 2(b). Figs. 2(c)-2(h) show ODF fields of a sample of 6 different subjects.

(a) ODF of mean after RAF (b) ODF of subject 1 after RAF

Figure 3. Anisotropic filtering on brain data. Figs. 3(a)-3(b) show

the ODF fields in Figs. 2(b) and 2(c) after applying RAF.

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